Order -Statistics

Question

The random variables $X_1, X_2, . . . , X_n, Y_1, Y_2, . . . , Y_n$ are iid $\mathcal{U}(0, a)$. Determine the distribution of $$Z_n = n \log\bigg(\frac{\max\{X(n), Y(n)\}}{\min\{X(n), Y(n)\}}\bigg)$$ Should I find the joint distribution of $\max$ and $\min$ and then find distribuation of $Z_n$, since we have a two different random variable I do not know how to do that!

Answer 1

First observe that the random vector $\Big(X_{(n)},Y_{(n)}\Big)$ is supported on $(0,a)^2$. Suppose $f$ is its joint density. Since $X_{(n)}$ and $Y_{(n)}$ are independent, we have $$f(x,y)=f_{X_{(n)}}(x)f_{Y_{(n)}}(y)=n^2a^{-2n}x^{n-1}y^{n-1}$$ for any $(x,y)\in (0,a)^2$. Also observe how $Z_n$ is supported on $[0,\infty)$, which means for any $z\geq 0$ we have $$F_{Z_n}(z)=P(Z_n\leq z)=P\Big(Z_n\leq z,X_{(n)}<Y_{(n)}\Big)+P\Big(Z_n\leq z,X_{(n)}\geq Y_{(n)}\Big)$$ With a little algebra we have $$F_{Z_n}(z)=P\Big(X_{(n)}<Y_{(n)}\leq e^{z/n}X_{(n)}\Big)+P\Big(Y_{(n)}\leq X_{(n)} \leq e^{z/n}Y_{(n)}\Big)$$ This probability can be written as the double integral $$F_{Z_n}(z)=2\int_0^{a} \int_{\frac{x}{e^{z/n}}}^{x}f(x,y)dydx=1-e^{-z}$$ which shows $Z_n \sim \text{Exp}(1)$.